Manuscript En05-B@1 · En05-B.tex · lemma · 8b626460

Lemma: Small periodic separation

The substrate: increasing directions and separated ports › An injective physical location map · lines 495–528 · proof included or linked

En05-B@1#lem:embedding, En05-B@1#eq:edgeLipschitz

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There are offsets $b_{p, i} \in R^{2}$ , with $∥ b_{p, i} ∥_{2} < ξ_{n}$ , such that

z_{n} (L_{k, p, i}) = z ˉ_{n} (L_{k, p, i}) + b_{p, i}

is injective on all sites, remains equivariant under hub translations, and satisfies

∥ z_{n} (x) - z_{n} (y) ∥_{2} \leq a_{n} for every graph edge x \sim y .

Proof. There are finitely many lane types $(p, i)$ . Choose their offsets successively in the open ball of radius $ξ_{n}$ . For a new type, equality modulo the lattice $δ_{n} Z^{2}$ with a hub or a previously chosen type excludes only a countable set of offsets. An open ball is not exhausted by that set. The resulting map is injective and translation equivariant.

A lane edge has length at most

\frac{ℓ _{n}}{N _{n}} + 2 ξ_{n} \leq \frac{ℓ _{n}}{N _{n}} + \frac{ℓ _{n}}{N _{n} ^{2}} = a_{n} .

All sites of $C_{k}$ are within $ξ_{n}$ of $δ_{n} k$ , except the hub itself, which is exactly there. A clique edge has length at most $2 ξ_{n} \leq a_{n}$ .

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En05-B@1#eq:edgeLipschitz · line 504
$∥ z_{n} (x) - z_{n} (y) ∥_{2} \leq a_{n} for every graph edge x \sim y .$
unlabeled · line 499
$z_{n} (L_{k, p, i}) = z ˉ_{n} (L_{k, p, i}) + b_{p, i}$
unlabeled · line 520
$\frac{ℓ _{n}}{N _{n}} + 2 ξ_{n} \leq \frac{ℓ _{n}}{N _{n}} + \frac{ℓ _{n}}{N _{n} ^{2}} = a_{n} .$

Lemma: Small periodic separation

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